Bickley–Naylor functions

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In physics, engineering, and applied mathematics, the Bickley–Naylor functions are a sequence of special functions arising in formulas for thermal radiation intensities in hot enclosures. The solutions are often quite complicated unless the problem is essentially one-dimensional[1] (such as the radiation field in a thin layer of gas between two parallel rectangular plates). These functions have practical applications in several engineering problems related to transport of thermal[2][3] or neutron,[4][5] radiation in systems with special symmetries (e.g. spherical or axial symmetry). W. G. Bickley was a British mathematician born in 1893.[6]

Definition

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The nth Bickley−Naylor function   is defined by

 

and it is classified as one of the generalized exponential integral functions.

All of the functions   for positive integer n are monotonously decreasing functions, because   is a decreasing function and   is a positive increasing function for  .

Properties

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The integral defining the function   generally cannot be evaluated analytically, but can be approximated to a desired accuracy with Riemann sums or other methods, taking the limit as a → 0 in the interval of integration, [aπ/2].

Alternative ways to define the function   include the integral,[7] integral forms the Bickley-Naylor function:

 
 
 
 
 

where   is the modified Bessel function of the zeroth order. Also by definition we have  .

Series expansions

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The series expansions of the first and second order Bickley functions are given by:

 
 

where γ is the Euler constant and

 

Recurrence relation

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The Bickley functions also satisfy the following recurrence relation:[8]

 

where  .

Asymptotic expansions

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The asymptotic expansions of Bickley functions are given as[9]

 
for  

Successive differentiation

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Differentiating   with respect to x gives

 

Successive differentiation yields

 

The values of these functions for different values of the argument x were often listed in tables of special functions in the era when numerical calculation of integrals was slow. A table that lists some approximate values of the three first functions Kin is shown below.

       
0 1.570796327 1.000000000 0.785398162
0.1 1.22863188 0.862521290 0.692543328
0.2 1.023679877 0.750458533 0.612064472
0.3 0.868832269 0.656147929 0.541862953
0.4 0.745203394 0.575660412 0.480375442
0.5 0.643693806 0.506373657 0.426358257
0.6 0.558890473 0.446366680 0.378791860
0.7 0.487198347 0.394159632 0.336825253
0.8 0.426061805 0.348575863 0.299739399
0.9 0.373578804 0.308659297 0.266921357
1.0 0.328286478 0.273620752 0.237845082
1.2 0.254888907 0.21564418 0.189162878
1.4 0.199050709 0.17049927 0.150734408
1.6 0.156156459 0.135163924 0.120310892
1.8 0.122960838 0.107392071 0.096165816
2.0 0.097120592 0.085490579 0.076963590
2.5 0.054422478 0.048670845 0.044307124
3.0 0.030848237 0.027924583 0.025646500
3.5 0.017634408 0.016117448 0.014909740
4.0 0.010146756 0.009346971 0.008698789
4.5 0.005868829 0.005441695 0.005090280
5.0 0.003408936 0.003178387 0.002986247
6.0 0.001161774 0.001092877 0.001034238
7.0 0.000400052 0.000378912 0.000360620
8.0 0.000138841 0.000132222 0.000126417
9.0 0.000048484 0.000046377 0.000044509
10 0.000017015 0.000016336 0.000015728

Computer code

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Computer code in Fortran is made available by Amos.[10]

See also

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References

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  1. ^ Michael F. Modest, Radiative Heat Transfer, p. 282, Elsevier Science 2003
  2. ^ Z. Altaç, Exact series expansions, recurrence relations, properties and integrals of the generalized exponential integral functions, Journal of Quantitative Spectroscopy & Radiative Transfer 104 (2007) 310–325
  3. ^ Z. Altaç, Integrals Involving Bickley and Bessel Functions in Radiative Transfer, and Generalized Exponential Integral Functions, J. Heat Transfer 118(3), 789−792 (August 1, 1996)
  4. ^ T. Boševski, An Improved Collision Probability Method for Thermal-Neutron-Flux Calculation in a Cylindrical Reactor Cell, NUCLEAR SCIENCE AND ENGINEERING:. 42, 23−27 (1970)
  5. ^ E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport, John Wiley Sons, 1984.
  6. ^ G. S. Marliss W. A. Murray, William G. Bickley—An appreciation, Comput J (1969) 12 (4): 301–302.
  7. ^ A. Baricz, T. K. Pogany, Functional Inequalities for the Bickley Function, Mathematical Inequalities and Applications, Volume 17, Number 3 (2014), 989–1003
  8. ^ M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, pp. 483, Dover Publications Inc., (1972).
  9. ^ M. S. Milgram, Analytic method for the numerical solution of the integral transport equation for a homogeneous cylinder, Nucl. Sci. Eng. 68, 249-269 (1978).
  10. ^ D. E. Amos, ALGORITH 609: A portable FORTRAN Subroutine for the Bickley Functions Kin(x), ACM Transactions on Mathematical Software, December 1983, 789−792